Maclaurin spheroid

A Maclaurin spheroid is an oblate spheroid which arises when a self-gravitating fluid body of uniform density rotates with a constant angular velocity. This spheroid is named after the Scottish mathematician Colin Maclaurin, who formulated it for the shape of Earth in 1742.^[1] In fact the figure of the Earth is far less oblate than Maclaurin's formula suggests, since the Earth is not homogeneous, but has a dense iron core. The Maclaurin spheroid is considered to be the simplest model of rotating ellipsoidal figures in hydrostatic equilibrium since it assumes uniform density.

Maclaurin formula[]

Angular velocity for Maclaurin spheroid

For a spheroid with equatorial semi-major axis $a$ and polar semi-minor axis $c$ , the angular velocity $\Omega$ about $c$ is given by Maclaurin's formula^[2]

{\frac {\Omega ^{2}}{\pi G\rho }}={\frac {2{\sqrt {1-e^{2}}}}{e^{3}}}(3-2e^{2})\sin ^{-1}e-{\frac {6}{e^{2}}}(1-e^{2}),\quad e^{2}=1-{\frac {c^{2}}{a^{2}}},

where $e$ is the eccentricity of meridional cross-sections of the spheroid, $\rho$ is the density and $G$ is the gravitational constant. The formula predicts two possible equilibrium figures when $\Omega \rightarrow 0$ , one is a sphere ( $e\rightarrow 0$ ) and the other is a very flattened spheroid ( $e\rightarrow 1$ ). The maximum angular velocity occurs at eccentricity $e=0.92996$ and its value is $\Omega ^{2}/(\pi G\rho )=0.449331$ , so that above this speed, no equilibrium figures exist. The angular momentum $L$ is

{\frac {L}{\sqrt {GM^{3}{\bar {a}}}}}={\frac {\sqrt {3}}{5}}\left({\frac {a}{\bar {a}}}\right)^{2}{\sqrt {\frac {\Omega ^{2}}{\pi G\rho }}}\ ,\quad {\bar {a}}=(a^{2}c)^{1/3}

where $M$ is the mass of the spheroid and ${\bar {a}}$ is the mean radius, the radius of a sphere of the same volume as the spheroid.

Stability[]

For a Maclaurin spheroid of eccentricity greater than 0.812670,^[3] a Jacobi ellipsoid of the same angular momentum has lower total energy. If such a spheroid is composed of a viscous fluid, and if it suffers a perturbation which breaks its rotational symmetry, then it will gradually elongate into the Jacobi ellipsoidal form, while dissipating its excess energy as heat. This is termed secular instability. However, for a similar spheroid composed of an inviscid fluid, the perturbation will merely result in an undamped oscillation. This is described as dynamic (or ordinary) stability.

A Maclaurin spheroid of eccentricity greater than 0.952887^[3] is dynamically unstable. Even if it is composed of an inviscid fluid and has no means of losing energy, a suitable perturbation will grow (at least initially) exponentially. Dynamic instability implies secular instability (and secular stability implies dynamic stability).^[4]

References[]

^ Maclaurin, Colin. A Treatise of Fluxions: In Two Books. 1. Vol. 1. Ruddimans, 1742.
^ Chandrasekhar, Subrahmanyan. Ellipsoidal figures of equilibrium. Vol. 10. New Haven: Yale University Press, 1969.
^ ^a ^b Poisson, Eric; Will, Clifford (2014). Gravity: Newtonian, Post-Newtonian, Relativistic. Cambridge University Press. pp. 102–104. ISBN 978-1107032866.
^ Lyttleton, Raymond Arthur (1953). The Stability Of Rotating Liquid Masses. Cambridge University Press. ISBN 9781316529911.

[1] Maclaurin, Colin. A Treatise of Fluxions: In Two Books. 1. Vol. 1. Ruddimans, 1742.

[2] Chandrasekhar, Subrahmanyan. Ellipsoidal figures of equilibrium. Vol. 10. New Haven: Yale University Press, 1969.

[PoissonAndWill-3] Poisson, Eric; Will, Clifford (2014). Gravity: Newtonian, Post-Newtonian, Relativistic. Cambridge University Press. pp. 102–104. ISBN 978-1107032866.

[Lyttleton1953-4] Lyttleton, Raymond Arthur (1953). The Stability Of Rotating Liquid Masses. Cambridge University Press. ISBN 9781316529911.

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Maclaurin spheroid

Contents

Maclaurin formula[]

Stability[]

See also[]

References[]